Pembrolizumab and Combination Chemotherapy in Treating Patients With Relapsed or Refractory Hodgkin Lymphoma

Inclusion Criteria: * Patients must have histologically confirmed diagnosis of classical Hodgkin lymphoma including nodular sclerosis, mixed cellularity, lymphocytic-rich, and lymphocyte depleted subtypes by the 4th edition of the World Health Organization (WHO) Classification of Tumors of Hematopoietic and Lymphoid Tissues published in 2008 * Patients must have disease with FDG-PET/CT avidity * Patients must have relapsed/refractory disease, with at least one line of prior chemotherapy, but =\< 2 prior lines of treatment, for Hodgkin lymphoma; NOTE: Patients must not have had prior immune checkpoint inhibitors; however, there are no other limitations to prior agent or regimen types * Patients must have Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Patients must have adequate organ and bone marrow function within 10 days of registration, as defined below: * Absolute neutrophil count \>= 1,000/mcL (in the absence of granulocyte colony stimulating factor (GCSF) for \>= 14 days) * Platelets \>= 75,000/mcl (in the absence of platelet transfusion for \>= 14 days) * Hemoglobin \>= 7g/dL (transfusion permitted) * Total bilirubin =\< 2 x institutional upper limit of normal (ULN); if total bilirubin is \> 2 x ULN, the direct bilirubin must be normal * Aspartate aminotransferases (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SPGT\]) =\< 2.5 x institutional ULN * Creatinine =\< 2 x ULN or creatinine clearance (CrCl) \> 30 ml/min * Female subject of childbearing potential should have a negative urine or serum pregnancy within 72 hours prior to registration; if the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Females of childbearing potential (FOCBP) should be willing to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 120 days after the last dose of study medication; NOTE: A FOCBP is any woman (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) who meets the following criteria: * Has not undergone a hysterectomy or bilateral oophorectomy * Has had menses at any time in the preceding 12 consecutive months (and therefore has not been naturally postmenopausal for \> 12 months) NOTE: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject * Male subjects should agree to use an adequate method of contraception starting with the first dose of study therapy through 120 days after the last dose of study therapy * Patients must have the ability to understand and the willingness to sign a written informed consent prior to registration on study Exclusion Criteria: * Patients who have had chemotherapy, radiotherapy, monoclonal antibody (mAb), or targeted small molecule therapy within 4 weeks of study registration are not eligible; those who have not recovered from adverse events (grade 1 or baseline) due to such agents administered more than 4 weeks earlier are not eligible * Patients may not be currently receiving any other investigational agents within 4 weeks of study registration * Patients must not have had prior exposure to any immune checkpoint inhibitors including anti-PD-1, anti-PD-L1 agents, anti-PD-L2 agents, or anti-CTLA-4 monoclonal antibodies * Patients must not have known central nervous system (CNS) involvement * Patients must not have had prior stem cell transplantation (autologous or allogeneic) * Patients must not have persistent diarrhea greater than National Cancer Institute (NCI) CTCAE grade 2 at the time of study registration, despite medical management * Patients must not have a history of (non-infectious) pneumonitis that required steroids, evidence of interstitial lung disease or active, noninfectious pneumonitis * Patients with known immunodeficiency, known autoimmune disease, or concurrent use of immunomodulatory agents including systemic steroids within 7 days prior to registration, are ineligible * Patients must not have co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens; this includes, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Patients must not have a known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer * Patients with known human immunodeficiency virus (HIV) infection or active TB (Bacillus tuberculosis) are not eligible * Patients with known active hepatitis B virus (HBV) or hepatitis C virus (HCV) infection are not eligible * Patients must not have a hypersensitivity to pembrolizumab or any of its excipients * Patients must not have received a live vaccine within 30 days of registration Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed) * Patients must not be pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment * Patients who are unwilling or unable to comply with the protocol or have known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial are not eligible